# Functions

Intuitively, a continuous function is a function whose graph can be drawn without lifting a pen. However, this notion must be made mathematically precise and is denoted in the following manner.

A real function ${\textstyle f: D \rightarrow \mathbb{R}}$ defined on some domain ${\textstyle D \subseteq \mathbb{R}}$ is continuous on a point ${\textstyle d \in D}$ if the following is satisfied

${\displaystyle \lim_{x \to d} f(x) = f(d).}$

This definition would have worked except that we run into the problem of having a neighbourhood of ${\textstyle d}$. Otherwise, how could one know if there are points sufficiently close to ${\textstyle d}$ in the domain of ${\textstyle f}$? To make this precise, there are the following two types of definitions.

## Continuous functions on an open interval

A function ${\textstyle f:(a,b) \rightarrow \mathbb{R}}$ is said to be continuous on the open interval ${\textstyle (a,b)}$ if the following is true for each point ${\textstyle d \in (a,b)}$.

${\displaystyle \lim_{x \to d} f(x) = f(d).}$